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March 12th, 2010, 09:34 AM   #1
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I can't understand this proof about Uniform Structures

There is a uniform structure U, defining a uniform topology associated with U. For every U U, U[x] is U evaluated at x, and U[A] is the union of U evaluated at every point in A.

The theorem is:
U[A] B for some U, then

The proof is below:
We may suppose that U is symmetric since there is a symmetric V such that V[A] (I don't understand how we can conclude that U is symmetric based on the fact that it has a symmetric subset; I understand the rest of the proof.)

(by symmetry of U)
(implies closure of A is in B)

I figured this out, instead of U, use the arguments on the symmetric V. V[a] is a subset of U[a].
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